ok finally got it working.
It looks like I was doing the “domain mapping” in the wrong place. I needed to do it on the dashboard of the “new” site. I then clicked to make it the primary site and all started working.
Is it possible to do the same with the results from a “search” – I have a search widget on the site. i.e. I want the results of a search to just show the titles and not the full articles.